# Elliptic Curve Scalar Field ## Metadata **Status**:: #x **Zettel**:: #zettel/fleeting **Created**:: [[2024-03-25]] **Parent**:: [[Elliptic Curve]] ## Synopsis Scalar field is used in the operations performed on the curve, such as point addition, scalar multiplication, and pairing. For elliptic curve over [[Finite Field]], adding a point G to itself repeatedly creates a cyclic group. The number of items in the group is the order of the point G. Attention that a cyclic group is also an abelian group. If pairing is defined for the cyclic group, they will constitute the scalar field. So the scalar field order is the order of the point G. When a point G on an elliptic curve over a [[Finite Field]] is repeatedly added to itself, it forms a finite abelian group. The total number of elements in this group is known as the order of the point G. The scalar field consists of the abelian group and a pairing operator. Therefore, the order of G also represents the order of the scalar field. See also [[Field Characteristic]].